Annina Bracher

Decoding Binary Node Labels from Censored Edge Measurements: Phase Transition and Efficient Recovery

We consider the problem of clustering a graph G into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y = B_Gx ⊕ Z, where B_G is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a uniform prior), and Z is a noise vector with Bernoulli (ε) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of x is possible if and only the graph G is connected, with a sharp threshold at the edge probability log (n)/n for Erdos-Renyi random graphs. The first goal of this paper is to determine how the edge probabilityp needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by α = np/log(n), it is shown that exact recovery is possible if and only if α > 2/(1 - 2ε)² + o(1/(1 - 2ε)²). In other words, 2/(1 - 2ε)² is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph G, defining the degree rate as α = d/log(n), where d is the minimum degree of the graph, it is shown that the proposed method achieves the rate α > 4((1 + λ)/(1 - λ)²/(1 - 2ε)² + o(1/(1 - 2ε)²), where 1-λ is the spectral gap of the graph G.

Probabilistic Recovery Guarantees for Sparsely Corrupted Signals

We consider the recovery of sparse signals subject to sparse interference, as introduced by Studer , IEEE T-IT, 2012. We present novel probabilistic recovery guarantees for this framework, covering varying degrees of knowledge of the signal and interference support, which are relevant for a large number of practical applications. Our results assume that the sparsifying dictionaries are characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.

Linear inverse problems on Erdős-Rényi graphs: Information-theoretic limits and efficient recovery

This paper considers the inverse problem with observed variables Y = BGX ⊕ Z, where BG is the incidence matrix of a graph G, X is the vector of unknown vertex variables with a uniform prior, and Z is a noise vector with Bernoulli(ε) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery of X is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n)=n for Erdös-Rényi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by α = np= log(n), it is shown that exact recovery is possible if and only if α > 2/(1-2ε)2+o(1/(1-2ε)2). In other words, 2/(1-2ε)2 is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. Full version available in [1].

Feedback and partial message side-information on the semideterministic broadcast channel

The capacity of the semideterministic discrete memoryless broadcast channel (SD-BC) with partial message side-information (P-MSI) at the receivers is established. It is shown that P-MSI to the deterministic receiver can only increase capacity if also the stochastic receiver has P-MSI, while P-MSI to the stochastic receiver alone can increase capacity. These capacity results are used to show that on the SD-BC with or without P-MSI feedback from the stochastic receiver can increase capacity; with P-MSI at the deterministic receiver it can, in particular, increase the sum-rate capacity. When the stochastic receiver has full MSI (F-MSI), feedback cannot increase capacity.

Distributed storage for data security

We study the secrecy of a distributed storage system for passwords. The encoder, Alice, observes a length-n password and describes it using two hints, which she then stores in different locations. The legitimate receiver, Bob, observes both hints. In one scenario we require that the number of guesses it takes Bob to guess the password approach 1 as n tends to infinity and in the other that the size of the list that Bob must form to guarantee that it contain the password approach 1. The eavesdropper, Eve, sees only one of the hints; Alice cannot control which. For each scenario we characterize the largest normalized (by n) exponent that we can guarantee for the number of guesses it takes Eve to guess the password.

Feedback, Cribbing, and Causal State Information on the Multiple-Access Channel

The benefits afforded by feedback and/or causal state information (SI) on the state-dependent discrete memoryless multiple-access channel (SD-MAC) with cribbing encoder/s are studied. Capacity regions are derived for communication scenarios whose capacities without cribbing are still unknown. It is shown that when the encoders can crib, the SD-MAC behaves less like a MAC and more like a single-user channel: 1) feedback does not help; 2) strictly causal SI does not help; and 3) causal SI to both encoders is best utilized using Shannon strategies. However, in asymmetric settings, the single-user-like behavior may or may not occur. For example, the SD-MAC with only one cribbing encoder is single-user-like when the state is revealed to the cribbing encoder, but not if it is revealed to the noncribbing encoder.

On feedback, cribbing, and causal state-information on the multiple-access channel

We show that the capacity region of the state-dependent multiple-access channel (SD-MAC) with strictly-causally cribbing encoders is not enlarged if strictly-causal state-information (SI) and feedback are furnished to the encoders. We also derive the capacity region of the SD-MAC with causal SI at the cribbing encoders and show that Shannon strategies are optimal. Such strategies are generally suboptimal if the encoders access distinct SI. However, Shannon strategies are optimal and we have a characterization of the capacity region for the case where both encoders crib, causal SI is revealed to one encoder, and feedback is available to the other encoder.

Coherence-based probabilistic recovery guarantees for sparsely corrupted signals

In this paper, we present novel probabilistic recovery guarantees for sparse signals subject to sparse interference, covering varying degrees of knowledge of the signal and interference support. Our results assume that the sparsifying dictionaries are characterized by coherence parameters and we require randomness only in the signal and/or interference. The obtained recovery guarantees show that one can recover sparsely corrupted signals with overwhelming probability, even if the sparsity of both the signal and interference scale (near) linearly with the number of measurements.

Guessing Attacks on Distributed-Storage Systems

We study the secrecy of a distributed-storage system for passwords. The encoder, Alice, observes a length-n password and describes it using δ s-bit hints, which she stores in different locations. The legitimate receiver, Bob, observes ν of those hints. In one scenario we require that the expected number of guesses it takes Bob to guess the password approach 1 as n tends to infinity, and in the other that the expected size of the shortest list that Bob must form to guarantee that it contain the password approach 1. The eavesdropper, Eve, sees η <; ν hints. Assuming that Alice cannot control which hints Bob and Eve observe, we characterize for each scenario the largest normalized (by n) exponent that we can guarantee for the expected number of guesses it takes Eve to guess the password.

The Zero-Error Feedback Capacity of State-Dependent Channels

The zero-error feedback capacity of the Gel’fand–Pinsker channel is established. It can be positive even if the channel’s zero-error capacity is zero in the absence of feedback. Moreover, the error-free transmission of a single bit may require more than one channel use. These phenomena do not occur when the state is revealed to the transmitter causally, a case that is solved here using Shannon strategies. Cost constraints on the channel inputs or channel states are also discussed, as is the scenario where—in addition to the message—also the state sequence must be recovered.